# Symbol-Based Analysis of Finite Element and Isogeometric B-Spline Discretizations of Eigenvalue Problems: Exposition and Review

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## Abstract

We present an example-based exposition and review of recent advances in symbol-based spectral analysis. We consider constant- and variable-coefficient, second-order eigenvalue problems discretized through the (isogeometric) Galerkin method based on B-splines of degree *p* and smoothness \(C^k\), \(0\le k\le p-1\). For each discretized problem, we compute the so-called symbol, which is a function describing the asymptotic singular value and eigenvalue distribution of the associated discretization matrices. Using the symbol, we are able to formulate analytical predictions for the eigenvalue errors occurring when the exact eigenvalues are approximated by the numerical eigenvalues. In this way, we recover and extend previous analytical spectral results. We are also able to predict the existence of \(p-k\) spectral branches, one “acoustical” and \(p-k-1\) “optical”, when discretizing the one-dimensional Laplacian eigenvalue problem. We provide explicit and implicit analytical expressions for these branches, and we quantify the divergence to infinity with respect to *p* of the largest optical branch in the case of \(C^0\) smoothness (the case of classical finite element analysis).

## Notes

### Acknowledgements

The authors express their gratitude to Emanuele Corti for his invaluable informatics support and for his help with the numerical experiments. C. Garoni is a Marie-Curie fellow of INdAM (Istituto Nazionale di Alta Matematica) under grant agreement PCOFUND-GA-2012-600198. H. Speleers was partially supported by the Mission Sustainability Program of the University of Rome Tor Vergata through the project “IDEAS”, and by INdAM Finanziamenti Premiali through the project “SUNRISE”. S-E. Ekström was supported by the FMB – Swedish National Graduate School of Mathematics and Computing. A. Reali was partially supported by Fondazione Cariplo – Regione Lombardia through the project “Verso nuovi strumenti di simulazione super veloci ed accurati basati sull’analisi isogeometrica”, within the program RST – rafforzamento. S. Serra-Capizzano was supported in part by the Donation KAW 2013.0341 from the Knut & Alice Wallenberg Foundation in collaboration with the Royal Swedish Academy of Sciences. T.J.R. Hughes was partially supported by the Office of Naval Research (Grant Nos. N00014-17-1-2119, N00014-17-1-2039, and N00014-13-1-0500), and by the Army Research Office (Grant No. W911NF-13-1-0220). C. Garoni, A. Reali, S. Serra-Capizzano, and H. Speleers are members of the INdAM Research group GNCS (Gruppo Nazionale per il Calcolo Scientifico).

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that there is no conflict of interest.

## References

- 1.Barbarino G (2017) Equivalence between GLT sequences and measurable functions. Linear Algebra Appl 529:397–412MathSciNetCrossRefGoogle Scholar
- 2.de Boor C (2001) A practical guide to splines, Revised edn. Springer, New YorkzbMATHGoogle Scholar
- 3.Böttcher A, Garoni C, Serra-Capizzano S (2017) Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey. Sb Math 208:1602–1627MathSciNetCrossRefGoogle Scholar
- 4.Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, ChichesterCrossRefGoogle Scholar
- 5.Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195:5257–5296MathSciNetCrossRefGoogle Scholar
- 6.Donatelli M, Garoni C, Manni C, Serra-Capizzano S, Speleers H (2016) Spectral analysis and spectral symbol of matrices in isogeometric collocation methods. Math Comp 85:1639–1680MathSciNetCrossRefGoogle Scholar
- 7.Ekström S-E, Furci I, Garoni C, Manni C, Serra-Capizzano S, Speleers H (2018) Are the eigenvalues of the B-spline isogeometric analysis approximation of \(-\Delta u=\lambda u\) known in almost closed form? Numer Linear Algebra Appl 25:e2198Google Scholar
- 8.Ekström S-E, Garoni C. A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices. Numer Algor https://doi.org/10.1007/s11075-018-0508-0
- 9.Garoni C (2018) Spectral distribution of PDE discretization matrices from isogeometric analysis: the case of \(L^1\) coefficients and non-regular geometry. J Spectral Theory 8:297–313MathSciNetCrossRefGoogle Scholar
- 10.Garoni C, Manni C, Pelosi F, Serra-Capizzano S, Speleers H (2014) On the spectrum of stiffness matrices arising from isogeometric analysis. Numer Math 127:751–799MathSciNetCrossRefGoogle Scholar
- 11.Garoni C, Manni C, Serra-Capizzano S, Sesana D, Speleers H (2017) Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods. Math Comp 86:1343–1373MathSciNetCrossRefGoogle Scholar
- 12.Garoni C, Manni C, Serra-Capizzano S, Sesana D, Speleers H (2017) Lusin theorem, GLT sequences and matrix computations: an application to the spectral analysis of PDE discretization matrices. J Math Anal Appl 446:365–382MathSciNetCrossRefGoogle Scholar
- 13.Garoni C, Mazza M, Serra-Capizzano S (2018) Block generalized locally Toeplitz sequences: from the theory to the applications. Axioms 7:49CrossRefGoogle Scholar
- 14.Garoni C, Serra-Capizzano S (2017) Generalized locally Toeplitz sequences: theory and applications (Volume I). Springer, ChamCrossRefGoogle Scholar
- 15.Garoni C, Serra-Capizzano S. Generalized locally Toeplitz sequences: theory and applications (Volume II). Springer (to appear)Google Scholar
- 16.Garoni C, Serra-Capizzano S, Sesana D (2015) Spectral analysis and spectral symbol of \(d\)-variate \(\mathbb{Q}_{\varvec {p}}\) Lagrangian FEM stiffness matrices. SIAM J Matrix Anal Appl 36:1100–1128MathSciNetCrossRefGoogle Scholar
- 17.Garoni C, Serra-Capizzano S, Sesana D. Block generalized locally Toeplitz sequences: topological construction, spectral distribution results, and star-algebra structure. To appear in a volume of the Springer INdAM SeriesGoogle Scholar
- 18.Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, New YorkzbMATHGoogle Scholar
- 19.Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefGoogle Scholar
- 20.Hughes TJR, Evans JA, Reali A (2014) Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput Methods Appl Mech Eng 272:290–320MathSciNetCrossRefGoogle Scholar
- 21.Hughes TJR, Reali A, Sangalli G (2008) Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of \(p\)-method finite elements with \(k\)-method NURBS. Comput Methods Appl Mech Eng 197:4104–4124MathSciNetCrossRefGoogle Scholar
- 22.Lyche T, Manni C, Speleers H (2018) Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In: Lyche T et al (eds) Splines and PDEs: from approximation theory to numerical linear algebra, Lecture Notes in Mathematics 2219, pp 1–76, Springer International PublishingGoogle Scholar
- 23.Reali A (2006) An isogeometric analysis approach for the study of structural vibrations. J Earthq Eng 10:1–30Google Scholar
- 24.Schumaker LL (2007) Spline functions: basic theory, 3rd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- 25.Serra-Capizzano S (2003) Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl 366:371–402MathSciNetCrossRefGoogle Scholar
- 26.Serra-Capizzano S (2006) The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl 419:180–233MathSciNetCrossRefGoogle Scholar
- 27.Tilli P (1998) Locally Toeplitz sequences: spectral properties and applications. Linear Algebra Appl 278:91–120MathSciNetCrossRefGoogle Scholar